Narrow Results

Let $X_{r,s}^n$ denote the blow-up of ${\mathbb{P}}^n$ along $r$ general lines and $s$ general points. In this paper, we focus on $l$-very ample line bundles on $X_{0,s}^n$ and investigate their Seshadri constants with some restrictions on $s$. Additionally, we compute the nef cone of $X_{r,0}^3$ for $r\le 6$ and study the Seshadri constants of some ample line bundles on it. We also examine the Seshadri constants of some ample line bundles on $X_{r,0}^4$$(r\le 7)$ and $X_{r,0}^5$$(r\le 5)$.

Following the formalism derived from one method of constructing common projective spaces along with using a special kind of odd module homomorphisms, denoted by ν, a novel supergeometric generalization of projective spaces is constructed. Existence of canonical line bundles over these spaces and their Chern classes are discussed.